TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 249, Number 1, April 1979 ON THE EXISTENCE OF NONREGULAR ULTRAFILTERS AND THE CARDINALITYOF ULTRAPOWERS BY MENACHEM MAGIDOR Abstract. Assuming the consistency of huge cardinals, we prove that <o3 can carry an ultrafilter D such that af'/D has cardinality <o3.(Hence D is not («3, w,) regular.) Similarly u>2can carry an ultrafilter D such that u"2/D has cardinality u2- (Hence D is not (<o2,w) regular.) 0. Introduction. When ultrafilters and ultrapowers (see [4]) were introduced, the problem of determining the cardinality of the ultrapower given the cardinality of the basis structure and of the index set became of interest (see [8]). As was already noted in [4], the problem can be completely settled, if one assumes that the ultrafilter in question is regular (see below for definition). Under the assumption of regularity the cardinality of the ultrapower has largest cardinality possible, i.e. if M is a structure of cardinality a > u and D is a uniform regular ultrafilter on an index set / of cardinality ß, then M'/D has cardinality otß. The problem whether every ultrafilter is regular, was posed by Keisler in [9], and partial answers were obtained in the constructible universe by Jensen and then improved by Prikry [14] who showed that if V = L then every uniform ultrafilter on un is regular. Results of Benda and Ketonen [2], Kanamori [6] and finally Ketonen [10] indicate the problem is closely connected with the existence of large cardinals. In particular (see [10]) if k + carries a uniform ultrafilter which is not (k+, k) regular, then 0* exists. On the other hand large cardinals immediately imply the existence of nonregular ultrafilters, see definition below. Note that a regular ultrafilter on k+ is (k+, k) regular though the converse does not necessarily hold. If D is an to, complete, nonprincipal ultrafilter on a cardinal k, then D is not regular: Note that K has to be at least as large as the first measurable in order to carry an co, complete ultrafilter. Keisler's problem is of course much more interesting for smaller cardinals like «,, w2 etc., and we attend this version of the problem. There is one fact known about the same problem for the continuum; if the continuum carries an w, saturated ideal then it carries a nonregular ultrafilter. Note that under this assumption the continuum is very large and this Received by the editors August 1, 1977 and, in revised form, September 26, 1977. AMS (MOS) subject classifications (1970). Primary 02K05; Secondary 02K35. Key words and phrases. Ultrafilter, ultraproduct, regular ultrafilter, huge cardinals. © 1979 American Mathematical Society 0002-9947/79/0000-01 53/$04.7S 97 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 98 MENACHEM MAGIDOR assumption is equiconsistent with the existence of measurable cardinals (see [16]). Our results, which naturally assume the existence of very large cardinals, show that already <o2can carry a nonregular ultrafilter and also that regu- larity is necessary for getting the cardinality results of [4] and [8]. In this paper we construct a model in which w2 carries an ultrafilter D such that for every countable structure, M, M°2/D has cardinality u2 which less than maximum possible (i.e. <o"2= 2"2). For «3 we get even a stronger result; the ultrafilter we get on <o3is not even (to3, w,) regular and if M is a structure having cardinality < <o, then A/"3//) has cardinality at most w3. Though we do not state this explicitly similar results hold for other cardinals. The w3 case also makes some progress on an old problem of Ulam. We know that if k is a successor cardinal k cannot cany a k complete, k saturated ideal, but assuming the consistency of huge cardinals, k can carry a k complete k+ saturated ideal. (Most of the ideas of §2 were derived from Kunen [11].) Naturally one wonders whether instead of relaxing the saturation requirement (i.e. the ideal is required to be k saturated), we can relax the completeness requirement (requiring of course that the ideal will be at least ul complete, otherwise the problem is trivial). The problem in this form was posed to us by S. Shelah. In the model we construct we can have an w3 saturated, w, complete uniform ideal on <o3.In the same model there is a family of <o30-1 measures defined on a field of subsets of <o3such that every subset of w3 is measurable with respect to at least one of the measures of the family. Compare this to 81 of [3] where the same problem is stated for to, instead of w3. 1. Preliminaries. Our set theoretic notation is standard (see [5]). For terminology concerning ultrafilters and ultrapowers see [1]. Lower case greek letters usually denote ordinals. The only exception being when they denote forcing terms. P(A) is the powerset of A, \A\ is the cardinality of A. If A is a. set of ordinals then A is the order type of A and PK{A) is the set of all subsets of A having ordertype k. An ultrafilter i/ona set / is uniform if for every A C I, A G U^\A\ = \I\. If |/| = \J\ then U naturally induces an equivalent ultrafilter on /, which we shall also denote by U and it should be clear from the context whether we refer to the ultrafilter on / or the corresponding ultrafilter on J. Let U be a filter over /, /, g are functions defined on /. We say that / is equivalent to g modulo U (/ =u g) if {/|i G I, g(0 = /(0) e U. The equiva- lence class of/with respect to = u will be denoted by [f]v. If (,Aa\a < y} and (Ba\a < X) are two partitions of / then (Aja < A) is equivalent to (Ba\a < X) modulo U ((Aa\a < X) =v (Aa\a < X}) if the function /, de- fined by /(0 = The only a such that / G Aa is equivalent modulo U to the License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE EXISTENCE OF NONREGULAR ULTRAFILTERS 99 function g defined by g(i) = The only a such that / G Ba. [(Aa\a < \}]u is the equivalence class of (Aja < A) with respect to =u- If U is a filter on / then A Ç I is of positive measure with respect to U if / - A G U. We drop "with respect to U" if the particular U we refer to is evident. Note that being of positive measure with respect to an ultrafilter simply means belonging to it. U is X complete if it is closed under inter- sections of length < X and U is À saturated if every family of sets of positive measure with respect to U such that the intersection of any two does not have positive measure with respect to U has cardinality < X. If / is a family of sets and U is a filter on /, then U is fine if for every a G U / {i\a G /'} G U. U is normal if for every choice function/on a set of positive measure A there is a set B C A of positive measure such that / is constant on B. (/ is a choice function if/(a) G a for every a in its domain.) If / is a family of subsets of X, and (J / = X then a fine filter U on / is called weakly normal if every choice function / on / is bounded with respect to U, i.e. there exists a < X such that {i\f(i) < a) G U. Note that neither normality nor weak normality implies the other property. An ultrafilter U on / is (a, X) regular if there are a members of U such that the intersection of any X of them is empty. U is regular if it is (|/|, co)regular. k is huge if there is a normal, fine, k complete ultrafilter on P"(A) for some A such that \A\ > k. An equivalent definition (see [7]) is that there exists an elementary embedding y of the universe V into some transitive class M such that j(a) = a for a < k, j(k) > k and M contains every subset of itself having cardinality < _/(/c). Note that if U is the normal, fine and k complete ultrafilter on P"(A) which appears in the definition of huge cardinal, then we can get the elementary embedding of the equivalent definition by forming the ultrapower Vp'w/ U and taking its transitive isomorph M, j the canonical embedding of V into VP'(A)/ U. In this ultrapower the identity function of P"(A) represents the set {j(a)\a < j(k)}. The reader is well aware that many set theoretic notions have different meanings in different universes. Usually it should be clear from the context in which universe the notion applies. In case of possible doubt we use superscript on our notation to indicate the universe in which it applies, thus PM(A) is the powerset of A in the sense of M, w,M is the ordinal which M considers to be w,, etc. Forcing will be very freely used in this paper, shifting between Boolean valued models and 0-1 models obtained by generic filters, as convenient. Since we shall iterate forcing constructions we use the following notation: If B is a complete Boolean algebra and T is a term which denotes a complete Boolean algebra in Vs then the iterated Boolean extension (VB)T can be considered to be one Boolean extension (see Solovay and Tennenbaum [17]).